A Strobe can drive you crazy. I can get it ALMOST stable, but as Dave points out, the very act of picking a note throws it off - the pick "stretches" the string as it passes through, which 1.) pulls the note initially slightly sharp, and 2.) changes the balance between the strings and the trem springs, which makes the trem pull the string back, oscillating slightly sharp initially.
I usually either pick SUPER lightly, or just brush the string with the tip of my thumb, and then wait until the note "settles in" to the sustain.
Also worth a try - the harmonic overtones can screw with the tuner a bit too. Try tuning on your neck pickup with the tone knob all the way down. It'll roll a lot of them out and give you a "purer" signal to work with.